James Calleja
©2015
Plan for the Session
Thinking about Student Agency & Responsibility
Task
Topic
Time
Introduction
What do you expect to get from today’s session? Aims of the session
¼ h
Working on a Task
Teachers work on the ‘Mathematics around a Giant chair’ task
½ h
Sharing reflections about your experiences working on this task
¼ h
Watch and discuss a video of Keith using this task with his Year 7 class
¾ h
Follow-‐up Reflection Lesson Video
A Teacher’s Narrative
Read and reflect on an excerpt from a teacher’s reflections as she implements changes to adopt more inquiry-‐based pedagogies.
½ h
Which issues do you find most striking? Why? Shifting Responsibility to Students: The Case of two Teachers
What actions might the teacher take to shift more instructional responsibilities onto the students?
The Didactical Contract
What is it? Why may it be important to discuss, share and establish with your students?
Which ones would you ‘risk’? How would you do that? With whom, when and why?
¾ h
½ h
Aims of the session For today’s session we will have the following aims: o To explore opportunities where the teacher may shift instructional responsibilities to the students o To reflect upon concerns in giving students more responsibilities over their learning o To understand the responsibilities and the roles of students and the teacher within a collaborative classroom community o To create an effective classroom culture based on habits, rules, expectations, behaviours, actions, interactions, beliefs and values which the teacher and the students establish, understand and share
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Teaching and Learning Mathematics through Inquiry
WORKING ON A MATHEMATICAL TASK
30 min
This is how you will work on the ‘Mathematics around a Giant chair’ task: • The task is presented to you
2 minutes
• You have some time to work individually generating your own questions about the displayed photo
3 minutes
• It is now time for you to work in groups of three or four to investigate a question of your choice. 15 minutes • You will present and share your ideas with the whole group
10 minutes
REFLECTING ON YOUR EXPERIENCE WORKING ON THE TASK
15 min
As a whole group you are asked to reflect on the following questions: 1. Comment on your experience working on this task. 2. Comment on the characteristics of the ‘Mathematics around a Giant chair’ task. 3. For which year group, do you think, would this task be suitable? Why? 4. What do you anticipate would be the challenges for students working on this task? How would you try to address these? 5. Would you consider doing this task with one of your classes next year? Why? 6. Would you present the task in a similar way as it was presented to you or would you do it differently? Why? How?
Teaching and Learning Mathematics through Inquiry
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The Giant Red Chair
! Student!Name:!________________________________! Students!I!am!working!with:!________________________________________________________________! ! Mathematical9related!questions:!(individually)!
Nehmen Sie Platz! !
! !
Question that we are going to answer is… !
!
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Teaching and Learning Mathematics through Inquiry
WATCHING A LESSON VIDEO
10 min
You will now watch a video of a teacher (Keith) using the ‘Mathematics around a Giant chair’ task with his Year 7 (form one) class of students. Note how the teacher structures the lesson, the mathematical ideas valued, the difficulties that students encounter and how the teacher deals with these issues. For the follow-‐up discussion, you are encouraged to write down some notes/points you might see as important. ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________
SOME POINTS TO THINK ABOUT
20 min
What are your comments about the lesson? Would you structure the lesson as the teacher did or would you do it differently? Who generates the mathematical ideas that get discussed? Who evaluates and/or responds to these ideas? How deeply do students get to explain their ideas? How does the teacher respond to students’ struggles? To what extent, do you think, the teacher stimulated student agency and shifted responsibility onto the students?
Teaching and Learning Mathematics through Inquiry
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IMPLEMENTING MORE INQUIRY-‐BASED PEDAGOGIES
30 min
The following extract is an account by Brea, a teacher of mathematics, who highlights changes in her thinking and practice as she shifts to implement more inquiry-‐based pedagogies. As you read underline, highlight or take notes on aspects that strike you in one way or another. These issues you highlight will guide our follow-‐up discussion. For most of my teaching career, I felt my job was to simplify mathematics. As a student of mathematics, I was led to believe that math should be simple, then when I started teaching, I thought that’s what my job was – to make math simple into little bits so the students could consume it and regurgitate it. So I aimed to cover the curriculum in consumable bits that could easily be delivered and tested. I planned to teach, not prepared to teach. Units were outcomes based, laid out in a day-‐by-‐day orderly manner. I delivered lessons with notes already compiled with set examples, complete with what pages and questions to do in the text. I did not realize how important the intellectual part of this job is and how I very easily could get or did get wrapped up in the skills and techniques of what to do in a classroom. Discourse has always been fundamental in my classroom, even when I wasn’t really working in inquiry. Conversation and dialogue has been the basis of my class. So the notion of relationship and conversations with kids was always there, but I never stepped outside of my preplanned boundary. In my classes, I would think kids were asking ‘good’ questions, but I now realize they were for clarification or procedural. We never critically entered a topic, looked at the bloodlines or cared for it in a way that honored it. If a student asked a question that seemed off topic or confusing to me, I would seldom really listen, often dismissing it. Even though we might discuss more than one way to a solution of an assigned problem, there was still a solution, that is, the problem was treated as closed. The focus of my efforts then would be on the students and building relationships, having conversations etc. At times I felt I was doing a good job because I was liked and I liked the kids also. Each semester brought newness in the form of students but the topics were set, flattened and I wondered how much longer I could do this. Since starting to engage in the inquiry kind of work that we are doing in my classroom, mathematics has become beautiful again. I want my students to understand that mathematics is not simple, that it is complex and complicated, that it does exist in the world, that it is a ‘living discipline’, that it has bloodlines. I want them to understand that there are patterns, but there are also no answers, there is no certainty. When they enter into the field, they are contributing in some way to it, but it is not meant to be simple and easy. I am finding that it is the structure of the mathematics and the patterns and the connections that seems to keep coming up as an entry point for me to be able to start to look at something to do with the kids. It is
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Teaching and Learning Mathematics through Inquiry
through its structure, patterns and connectedness I can see many possibilities. Where does this come from? Why do we still talk about it? How does it live and contribute to the world today? There are times I see clearly the mathematical connections either through the structure of math, its beauty, complexity or imagery. The world has opened up and through discourse math presents itself as complicated, uncertain, and unfinished. It is no longer, as tends to be in the math classroom, certain, linear, and algorithmic. I have begun to see more connections within topics and in interdisciplinary ways. The more we enter into a topic, the more exciting it becomes, it all seems new to me again, it is exciting and alive. But there are challenges. For example: How do I open topics in a generous way? Do I look for the topic in the world or see the world through the topic? Discourse continues to be fundamental in my classroom. Topics always open up with conversation. Students are always in partner or groups talking. They are always writing and sharing in some way, so that their work is always public in some form. But unlike the pre-‐inquiry classroom, the world has now opened up in the discourse and conversations are rich and complicated, answers are uncertain, the work constantly unfinished. I now want students to question, and wonder, and ask why. I want them to make connections and to see things as interconnected. I am also now deeply trying to listen to their inquiries. There are now portals in my lessons that call me to really listen, become attuned to what students are wondering about. They are wondering about math and are inquiring into topics that come up in class. For example, we were talking about the names of polynomial functions with a degree of one to five. A student asked, ‘‘what about 6, 7 etc.?’’ He was assigned the task of finding out about this for us all. The next class he said he could not look it up last night but four others responded that they had. What I have noticed of late is the openness of my students to think and go places they have not before. As I open a topic, I never know where it will go. More often than not, we end up in territory way beyond the ‘curriculum’ for that grade. For example, the grade 10’s, in a conversation about the sine and cosine of supplementary angles, ended up describing the unit circle. In an assignment in which they researched the life of a mathematician, they then wrote about how they could come to understand who they were and who they could become. It seemed natural to discuss these things. Yet I know if I had tried this before, I would not have had the open reception or the effort they put into their writing. I am constantly amazed at their thoughtfulness; at times they seem so much smarter than I. I truly feel privileged to be in the face of the young. The extract is taken from Chapman and Heater (2010, p. 450-‐451) Chapman, O., & Heater, B. (2010). Understanding change through a high school mathematics teacher’s journey to inquiry-‐based teaching. Journal of Mathematics Teacher Education, 13(6), 445-‐458.
Teaching and Learning Mathematics through Inquiry
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SHIFTING MORE RESPONSIBILITY TO STUDENTS A CASE OF TWO TEACHERS
40 min
In most classrooms, it seems that the teacher carries much of the responsibilities for student learning. And rightly so, some might claim. However, teachers seem to undertake full responsibility for whatever goes on in the classroom, with students ‘passively’ waiting for things to be done – by the teacher, for the students – because that’s the way it is and that’s the way it should be! Let’s consider some key decisions, actions and expectations traditionally undertaken by teachers. Maria and Helen are secondary school teachers and have taught mathematics for over five years. Both Maria and Helen feel that they work under constant pressures and constraints mainly related to the mathematical content that they are required to teach, the time factor and high-‐stakes examinations. Moreover, they feel that they also carry much of the responsibility for student learning. The statements that follow are taken from a conversation that Maria and Helen had regarding their classroom practices, their role as teachers and that of their students as learners of mathematics. Issue
Maria
Who decides on the I decide which exercises and work that students problems my students should do? do.
I provide a list of exercises but then allow my students to select which problems or questions to do.
How much work should I expect students to do?
I expect and make sure that all students do all the work that I assign.
I allow some degree of freedom with the amount of work students do.
Who corrects the students’ work?
I always collect and correct students’ work on class tasks and homeworks.
My students usually get to correct their own work and only get to hand it in when they cannot sort out problems on their own.
What if students have issues that they cannot solve?
I make sure that, by the end of the lesson, I sort out students’ unresolved mathematical issue.
I prefer to leave my students to struggle with their unresolved mathematical issues.
What about notes taking?
I make sure to give students my own set of notes about each topic.
I expect my students to write their own notes about the topic being done in class.
How do I set I always choose and decide students to work in with whom they get to work. groups?
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Helen
Teaching and Learning Mathematics through Inquiry
I provide students with the opportunity to choose and decide with whom to work.
ESTABLISHING A DIDACTICAL CONTRACT WITH STUDENTS
30 min
When students are given the task of deciding how much they need to do to understand the mathematics involved, an important shift takes place: away from doing something because they have been told to do so, and towards doing something because they recognize the value of making progress. Ollerton (2006, p. 198) References: Ollerton, M. (2006). Getting the Buggers to Add Up (2nd Edition). London: Continuum.
Pair work Using the above quote as our guiding principle, think about and identify how you may negotiate new ways of working with the students in your classroom. Some questions to help you focus: How would you communicate the classroom rules? Which rules do you value most? Why? How do you expect your students to work in your class? What habits and behaviours would you like to instill in the students? How do you intend to get students to accept more responsibility for their learning?
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Teaching and Learning Mathematics through Inquiry
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SESSION EVALUATION
10 min
Ø Briefly describe your experience during today’s session. ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ Ø What did you feel un/comfortable doing during the session? Comfortable: ___________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ Uncomfortable: ________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ Ø I used to think... but now I know… I used to think __________________________________________________________________________________ ___________________________________________________________________________________________________ Now I know ____________________________________________________________________________________ ___________________________________________________________________________________________________ Ø What will you take with you and try to implement in your class? ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ Ø Any other comments/suggestions that you would like to add. ___________________________________________________________________________________________________ ___________________________________________________________________________________________________ Thank you for your participation and reflections.
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Teaching and Learning Mathematics through Inquiry